The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This essential trains for: SAT-II, AMC-8, AMC-10, Math Kangaroo 9-10.

The Law of Sines

The area of a scalene triangle ABC can be computed using two sides and the angle between them:

Consider the triangle with side lengths a, b, c and angle B formed by the sides a and c. If we could compute the height corresponding to side c from this information, we could use the previous formula to find the area.

figure

We can achieve this by using the definition of the sine ratio:

equation

Solving for H:

equation

we can substitute this expression for H in:

equation

equation

Repeating the same procedure with other pairs of sides:

equation

Multiplying all the ratios by 2:

equation

Dividing by the product of all side lengths:

equation

where from we get:

equation

an identity which is also known as the law of sines.

The law of sines is useful when the data provided in the problem consists of a side length and measures of the two angles adjacent to it.

The Law of Cosines

Consider a scalene triangle in which we know the lengths of two of the sides and the measure of the angle adjacent to them.

If we are interested in computing the length of the third side, we can use an additional construction. In the triangle ABC draw the altitude AD:

figure

The goal is to find the length of AC (b). We cannot apply the Pythagorean theorem in a scalene triangle, but we can apply it twice in the two right angle triangles ABD and ADC.

Denote:

equation

Use the Pythagorean theorem in the triangle ABD:

equation

Use the definition of the cosine ratio to replace BD:

equation

equation

equation

Use the Pythagorean theorem in the triangle ADC:

equation

Use the definition of the cosine ratio to replace CD:

equation

equation

Now set the expressions for h2 to be equal:

equation

and do the algebra:

equation

equation

equation

Re-arranging the terms we get:

equation

which is known as the law of cosines.

By permutations, the following are also true:

equation

equation